For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), ...

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13
votes
3answers
835 views

How to show determinant of a specific matrix is nonnegative

How to show that $$\det A= \det \begin{pmatrix}\cos\frac{\pi}{n}&-\frac{\cos\theta_1}{2}&0&0&\cdots&0&-\frac{\cos\theta_n}{2} ...
13
votes
2answers
2k views

Topology of matrices

1.Consider the set of all $n×n$ matrices with real entries as the space $\mathbb R^{n^2}$ . Which of the following sets are compact? (a) The set of all orthogonal matrices. (b) The set of all ...
13
votes
3answers
417 views

when does $\det(AB^T+BA^T)\le \det(AA^T+BB^T)$ hold?

When does the following matrix inequality hold? $$\det(AB+B^TA^T)\le \det(AA^T+BB^T)$$ $A$ and $B$ are any real matrices. My reply gives a counter example. The question is under what condition ...
12
votes
2answers
666 views

What's the name for the property of a function $f$ that means $f(f(x))=x$?

I can think of several examples of functions such that twice application of the function is equivalent to no application of it. Additive inverse Multiplicative inverse Fourier transform Complex ...
12
votes
6answers
886 views

A matrix satisfying $AB-BA=B$

If $A$ and $B$ are two matrices of $\mathcal{M}_n(\mathbb{R}$) such that $$AB-BA=B$$ how can we prove that $B$ isn't invertible? my attempt: I found that $\mathrm{tr}(B)=0$ but I know that this is ...
12
votes
3answers
2k views

Why do the $n \times n$ non-singular matrices form an “open” set?

Why is the set of $n\times n$ real, non-singular matrices an  open subset of the set of all $n\times n$ real matrices? I don't quite understand what "open" means in this context. Thank you.
12
votes
5answers
298 views

Prove that $A^k = 0 $ iff $A^2 = 0$

Let $A$ be a $ 2 \times 2 $ matrix and a positive integer $k \geq 2$. Prove that $A^k = 0 $ iff $A^2 = 0$. I can make it to do this exercise if I have $ \det (A^k) = (\det A)^k $. But this ...
12
votes
4answers
4k views

How to calculate the matrix exponential explicitly for a matrix which isn't diagonalizable?

How can I compute an expression for $(\exp(Qt))_{i,j}$ for some fixed $i, j$ and matrix $Q$? When $Q$ is diagonalizable, we can diagonalize, but what can be done otherwise? Thanks.
12
votes
2answers
1k views

$AB-BA=I$ having no solutions

The question is from Artin's Algebra. If $A$ and $B$ are two square matrices with real entries, show that $AB-BA=I$ has no solutions. I have no idea on how to tackle this question. I tried block ...
12
votes
3answers
214 views

If $A$ is positive definite, then $\int_{\mathbb{R}^n}\mathrm{e}^{-\langle Ax,x\rangle}\text{d}x=\left|\det\left({\pi}^{-1}A\right)\right|^{-1/2}$

Let $A$ be a positive definite real $n\times n$ matrix. How can I prove that $$ \int_{\mathbb{R}^n}\mathrm{e}^{-\langle ...
12
votes
7answers
4k views

Relation between Cholesky and SVD

When we have a symmetric matrix $A = LL^*$, we can obtain L using Cholesky decomposition of $A$ ($L^*$ is $L$ transposed). Can anyone tell me how we can get this same $L$ using SVD or Eigen ...
12
votes
2answers
383 views

Geometric interpretation of the cofactor expansion theorem

I find the geometric interpretation of determinants to be really intuitive - they are the "area" created by the column vectors of the matrix. Could someone give me a geometric interpretation of the ...
12
votes
2answers
586 views

Why is the identity the only symmetric $0$-$1$ matrix with all eigenvalues positive?

While thinking about this question I managed to convince myself that the identity is the only symmetric $0$-$1$ matrix with all eigenvalues positive. However, the argument is rather low-level. It ...
12
votes
2answers
527 views

How many $n\times m$ binary matrices are there, up to row and column permutations?

I'm interested in the number of binary matrices of a given size that are distinct with regard to row and column permutations. If $\sim$ is the equivalence relation on $n\times m$ binary matrices such ...
12
votes
2answers
1k views

Integral of matrix exponential

Let $A$ be an $n \times n$ matrix. Then the solution of the initial value problem \begin{align*} \dot{x}(t) = A x(t), \quad x(0) = x_0 \end{align*} is given by $x(t) = \mathrm{e}^{At} x_0$. I am ...
12
votes
3answers
528 views

Let $A$ and $B$ be $n \times n$ real matrices such that $AB=BA=0$ and $A+B$ is invertible

I came across the following problem that says: Let $A$ and $B$ be $n \times n$ real matrices such that $AB=BA=0$ and $A+B$ is invertible. Then how can I prove the following: rank $A$+ ...
12
votes
1answer
3k views

Is it faster to multiply a matrix by its transpose than ordinary matrix multiplication?

I'm writing a program that multiples a matrix by its transpose, and was trying to find efficiency hacks I could exploit considering that the two matrices being multiplied are related. Any ideas?
12
votes
2answers
620 views

Properties of 4 by 4 Matrices

Define $ A=\begin{pmatrix} x_1 & x_2 & 0 & 0\\ 0 &1&0&0\\ 0&0&1&0\\ 0&0&0&1 \end{pmatrix}, B=\begin{pmatrix} 1 & 0 & 0 & 0\\ x_3 ...
12
votes
1answer
168 views

Compute $\det(A^n+B^n)$

Let $A, B $ be two real $3\times 3 $ matrices, $AB=BA$, and $ \det(A-B)=\det(A^2+B^2)=1,\det(A+B)=3, \det(B)=0 $, then, what is ? $$\det(A^n+B^n)$$ here $n$ is a positive integer. The problem ...
12
votes
1answer
773 views

Direct proof that nilpotent matrix has zero trace

Does anyone know a proof from the first principles that a nilpotent matrix has zero trace. No eigenvalues, no characteristic polynomials, just definition and basic facts about bases and matrices.
12
votes
2answers
395 views

Product of nilpotent matrices.

Let $A$ and $B$ be $n \times n$ complex matrices and let $[A,B] = AB - BA$. Question: If $A , B$ and $[A,B]$ are all nilpotent matrices, is it necessarily true that $\operatorname{trace}(AB) ...
12
votes
4answers
3k views

Square root of a matrix

Under what conditions does a matrix $A$ have a square root? I saw somewhere that this is true for Hermitian positive definite matrices(whose definition I just looked up). Moreover, is it possible ...
12
votes
1answer
109 views

Check membership in a matrix group

I'm looking for a (preferably somewhat efficient) algorithm for this problem: Given a normal subgroup of $SL(m, \mathbb{Z})$ generated by a finite set $\{M_1, M_2, \dotsc, M_n\}$, and some $A \in ...
12
votes
2answers
510 views

The infinite-dimensional limit of sequence of solutions of linear equations when the number of equations goes to infinity

Suppose we have an infinite-dimensional real vector $y=(y_1,...)$. Suppose we have an infinite-dimensional real matrix $C=(c_{ij})$, $i,j\in\mathbb{N}$. Let $C^k$ be a submatrix of $C$, ...
12
votes
3answers
272 views

Prove the inequality: $\prod_{j=1}^ka_{jj}\leq\left(\frac{1}{k}\sum_{j=1}^k\lambda_j\right)^k.$

To all those who are eagerly awaiting a new question, all those who love math, I give this challenge and I hope for you good moments of reflection. Let $A=(a_{ij})_n$ a real nonnegative symmetric ...
12
votes
2answers
652 views

A weak converse of $AB=BA\implies e^Ae^B=e^Be^A$ from “Topics in Matrix Analysis” for matrices of algebraic numbers.

It is a well known fact that if $A,B\in M_{n\times n}(\mathbb C)$ and $AB=BA$, then $e^Ae^B=e^Be^A.$ The converse does not hold. Horn and Johnson give the following example in their Topics in Matrix ...
12
votes
1answer
429 views

Eigenvalues of $A$ and $A + A^T$

This question has popped up at me several times in my research in differential equations and other areas: Let $A$ be a real $N \times N$ matrix. How are the eigenvalues of $A$ and $A + A^T$ related? ...
12
votes
0answers
152 views

determinant of a standard magic square

What is the lowest positive, what the highest possible value for the determinant of a standard-magic-square-matrix of order n ? Are there singular standard-magic-square-matrices of any order ...
12
votes
1answer
329 views

A sharper bound for $\|\cos(kA)\|_{\infty}$ for symmetric stochastic matrices

Given $A \in \mathbb{R}^{n \times n}$ that is symmetric, stochastic and diagonalizable, and $k \in \mathbb{N}$, I am interested in bounding $\|\cos(kA)\|_{\infty}$ from above. $\| \|_{\infty}$ is ...
12
votes
1answer
210 views

Characterization of Volumes of Lattice Cubes

Here is a problem that came up in a conversation with a professor. I do not know if he knew the answer (and told me none of it) and has since passed so I can no longer ask him about it. Let $C$ be a ...
12
votes
0answers
180 views

Fundamental domain of $\operatorname{GL}(n,\mathbb R)$ acted on by $\operatorname{GL}(n, \mathbb Z)$

What is a simple description of a fundamental domain of $\operatorname{GL}(n,\mathbb R)$ acted on by $\operatorname{GL}(n,\mathbb Z)$? $\operatorname{GL}(n,\mathbb R)$ is the group of all real ...
12
votes
3answers
573 views

Why is an orthogonal matrix called orthogonal?

I know a square matrix is called orthogonal if its rows (and columns) are pairwise orthonormal But is there a deeper reason for this, or is it only an historical reason? I find it is very confusing ...
11
votes
6answers
1k views

Matrix to power $2012$

How to calculate $A^{2012}$? $A = \left[\begin{array}{ccc}3&-1&-2\\2&0&-2\\2&-1&-1\end{array}\right]$ How can one calculate this? It must be tricky or something, cause there ...
11
votes
5answers
324 views

Matrices with $A^3+B^3=C^3$

Problem: Find infinitely many triples of nonzero $3\times 3$ matrices $(A,B,C)$ over the nonnegative integers with $$A^3+B^3=C^3.$$ My proposed solution is in the answers.
11
votes
3answers
860 views

Determinant of a specific circulant matrix, $A_n$

Let $$A_2 = \left[ \begin{array}{cc} 0 & 1\\ 1 & 0 \end{array}\right]$$ $$A_3 = \left[ \begin{array}{ccc} 0 & 1 & 1\\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{array}\right]$$ ...
11
votes
3answers
3k views

Rank of skew-symmetric matrix

Is there a succinct proof for the fact that the rank of a non-zero skew-symmetric matrix ($A = -A^T$) is at least 2? I can think of a proof by contradiction: Assume rank is 1. Then you express all ...
11
votes
6answers
363 views

Given a matrix $A$, find $A^n$

Given the matrix $$A = \left[{9\atop20}{-4\atop-9}\right]$$ how do I find $A^7$ or $A^{54}$ or $A^{2008}$ (etc.) ? I know I need the eigenvalues of A, but I'm not sure what to do afterwards. Is the ...
11
votes
3answers
2k views

$I_m - AB$ is invertible if and only if $I_n - BA$ is invertible.

The problem is from Algebra by Artin, Chapter 1, Miscellaneous Problems, Exercise 8. I have been trying for a long time now to solve it but I am unsuccessful. Let $A,B$ be $m \times n$ and $n ...
11
votes
1answer
5k views

Are the eigenvalues of $AB$ equal to the eigenvalues of $BA$? (Citation needed!)

First of all, am I being crazy in thinking that if $\lambda$ is an eigenvalue of $AB$, where $A$ and $B$ are both $N \times N$ matrices (not necessarily invertible), then $\lambda$ is also an ...
11
votes
2answers
505 views

Characteristic polynomials exhaust all monic polynomials?

Let $A$ be an $n\times n$ matrix, then $\mathrm{char}_A(x):=\det(xI-A)$ is a monic polynomial of degree $n$. It is called the characteristic polynomial of $A$. My question is the converse: Let ...
11
votes
4answers
585 views

What makes elementary row operations “special”?

This is probably a stupid question, but what makes the three magical elementary row operations, as taught in elementary linear algebra courses, special? In other words, in what way are they "natural" ...
11
votes
6answers
292 views

Maximizing the sum $\sum\limits_{i=1}^nx_ix_{i+1}$ subject to $\sum\limits_{i=1}^nx_i=0$ and $\sum\limits_{i=1}^nx_i^2=1$

Is there an efficient way of solving the following problem? Given $x_i\in \mathbb R$, and that $\sum\limits_{i=1}^nx_i=0$ and $\sum\limits_{i=1}^nx_i^2=1$. I want to maximize ...
11
votes
3answers
2k views

Recovering the two SU(2) matrices from SO(4) matrix

Since there is a 2-1 homomorphism from SU(2)xSU(2) to SO(4) there should be a way to recover the two SU(2) matrices given an SO(4) matrix. I believe I could set this up as a system of equations ...
11
votes
3answers
5k views

Eigenvalues for the rank one matrix $uv^T$

Suppose $A=uv^T$ where $u$ and $v$ are non-zero column vectors in ${\mathbb R}^n$, $n\geq 3$. $\lambda=0$ is an eigenvalue of $A$ since $A$ is not of full rank. $\lambda=v^Tu$ is also an eigenvalue of ...
11
votes
4answers
355 views

Finding all matrices $B$ such that $AB=BA$ for a fixed matrix $A$

Let $$ A=\begin{pmatrix} 1 & 0 & 0 \\ 0& 1 & 0 \\ 3 & 1 & 2 \end{pmatrix} $$ Find all matrices $B$ such that $AB=BA$. Attempt at solution: I can show that $A$ is ...
11
votes
3answers
304 views

Eigenvalue test faster than $O\left(n^3\right)$?

Given a real $n\times n$ matrix $A$, one can find the eigenvalues in $O\left(n^3\right)$ by using say, the $QR$ algorithm. Now, what if we guess an eigenvalue $\lambda_0$, and we want to know if it's ...
11
votes
2answers
762 views

Why are the eigenvalues of these “bitwise XOR matrices” integers?

In the course of playing around with this question, I have hit upon a question of my own. Consider the $n\times n$ symmetric matrix $\mathbf X$ whose entries are given by ...
11
votes
3answers
418 views

Eigenvalues of some peculiar matrices

While I was toying around with matrices, I chanced upon a family of tridiagonal matrices $M_n$ that take the following form: the superdiagonal entries are all $1$'s, the diagonal entries take the form ...
11
votes
5answers
325 views

Why are there$ 736$ $2\times 2$ matrices $(M)$ over $\mathbb{Z}_{26}$ for which it holds that $M=M^{-1}$?

I'm currently trying to introduce myself to cryptography. I'm reading about the Hill Cipher currently in the book Applied Abstract Algebra. The Hill Cipher uses an invertible matrix $M$ for ...
11
votes
2answers
1k views

Block Diagonal Matrix Diagonalizable

I am trying to prove that: The matrix $C = \left(\begin{smallmatrix}A& 0\\0 & B\end{smallmatrix}\right)$ is diaginalizable, if only if $A$ and $B$ are diagonalizable. If $A\in\mathbb{C}^n$ ...